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Lesson 5--Representations of angular dependence

This reading concerns the representation of current and flux angular dependence with approximate polynomials.  In the practice of nuclear engineering, however, the most common use of angular polynomial approximations is in the representation of scattering cross section distributions.  That is, if we have a scattering-type interaction of some particle with a stationary nucleus, the particle will deflect by some angle wpe42.gif (1043 bytes) :

wpe24.gif (2536 bytes)

Nuclear cross section experimentalists will develop data for particular particles, target nucleus, and particle energy.  The resulting function then must be expressed as some function of deflection angle; the method generally used is to present the data, wpe4D.gif (991 bytes) , in the form of Legendre expansion coefficients in the cosine, wpe4B.gif (1151 bytes) , using the approximation:

wpe4E.gif (1550 bytes)

The wpe4F.gif (1003 bytes) are the Legendre Polynomials that are discussed in Appendix B.4, which you read.

The Legendre polynomials are just fancy combinations of the regular algebraic function series wpe50.gif (1134 bytes) .   In that case, you might ask, why bother?  Why not just present the data as a "straight" algebraic expansion:

wpe51.gif (1307 bytes) ?



The answer is that the Legendre polynomials offer several useful advantages:

    The expansion coefficients can be found with a simple integration:
wpe52.gif (1448 bytes)
    The resulting expansion is equivalent to a least squares fit.

    The user can use a partial fit if desired by just truncating the series.

The last point is the most important, so let me elaborate on it.  If the data is fit to a Legendre expansion to, say, 5th order:

wpe56.gif (2590 bytes)

and the user decided that the current application only required a 2nd order expansion, the user could just use the first 3 coefficients, i.e.,

wpe57.gif (1829 bytes)

This is NOT true for the algebraic series.  One must use the whole series every time, which can unnecessarily add to the time and expense of an analysis.

Example:

An EXCEL spreadsheet has been developed that performs a P0-P5 fit of a user-specified function. It calculates the coefficients using an S16 Gaussian integration procedure. 

The resulting graph for a function:

wpe53.gif (1460 bytes)

is 

wpe54.gif (8046 bytes)


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